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Proof step 24 from proof prepost-remove (previous step)

ar1[m] = elemindex(a, m1), kar[m1] = key-index(key, m), ar = ar1[bh.ar[n].idx, ar1[bh.ar[n].idx].idx:= n], ar0 = ar1, bh = bh1.ar:= bh1.ar[n, bh1.ar[bh1.size]], bh0 = bin-heap(kar, m0), bh1 = bh0.size:= (bh0.size - 1), n = m1, 2exp-size(# kar), bh.ar[n].key < bh.ar[(n - 1) / 2].key, n < # bh.ar, bh.ar[n].idx < # ar1, n < bh0.size, n < # bh1.ar, bh1.size < # bh1.ar, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar1, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar, bh0.size ≥ 1, n ≠ 0 → n < # bh.ar ∧ (n - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n ≥ 1 ⊦ ⟪bin_min_heap_bubble_up#(n; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar0, ar) ∧ keyset(bh.ar, bh.size) = keyset(kar, m0) -- key-index(key, m)), # keyset(kar, m0) = 0, bh1.size = 0, bh1.size = n, keyset(kar, m0) = ∅, n = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
ar1[m] = elemindex(a, m1), kar[m1] = key-index(key, m), ar = ar1[kar[m0 - 1].idx, ar1[kar[m0 - 1].idx].idx:= m1], bh = bin-heap(kar[m1, kar[m0 - 1]], m0 - 1), # keyset(kar, m0) ≠ 0, # kar ≠ 1, m0 - 1 ≠ m1, keyset(kar, m0) ≠ ∅, m0 ≠ 1, m1 ≠ 0, 2exp-size(# kar), (m1 - 1) / 2 < # kar, kar[m0 - 1].key < kar[m1, kar[m0 - 1]][(m1 - 1) / 2].key, kar[m0 - 1].idx < # ar1, m1 < m0, m1 < # kar, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar1, m0), key-index(key, m) ∈ keyset(kar, m0), 1 ≤ m1, m0 ≤ # kar ⊦ ⟪bin_min_heap_bubble_up#(m1; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar1, ar) ∧ keyset(bh.ar, bh.size) = keyset(kar, m0) -- key-index(key, m))

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat% ⊦ m < n → n ≠ 0; (s)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ keyset(kar, 0) = ∅; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)%m < # kar ⊦ kar[m, ki][m] = ki; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m ≤ n → (n - m = 0 ↔ n = m); (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ bin-heap(kar, m).ar:= kar0 = bin-heap(kar0, m); (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ bin-heap(kar, m).size:= m0 = bin-heap(kar, m0); (s)% ⊦ bin-heap(kar, m).size = m; (s)